Original article iq.unesp.br/ecletica | Vol. 44 | n. 3 | 2019 | 50 Eclética Química Journal, vol. 44, n. 3, 2019, 50-55 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v44.3.2019.p50-55 Solution of the Schrödinger equation with inversely quadratic Yukawa potential (IQYP) plus Kratzer-Fues (KFP) potential using the WKB quantum mechanical formalism Benedict Iserom Ita , Hitler Louis+ , Nelson Nzeata-Ibe University of Calabar (Unical), Faculty of Physical Sciences, P.M.B 1115, Calabar, Cross River State, Nigeria. +Corresponding author: Hitler Louis, email address: louismuzong@gmail.com ARTICLE INFO Article history: Received: December 18, 2018 Accepted: February 25, 2019 Published: July 4, 2019 Keywords: 1. Schrödinger equation 2. inversely quadratic Yukawa potential 3. attractive Coulomb potential 4. Kratzer Fues potential 5. WKB approximation 1. Introduction One of the interesting problems in quantum mechanics is to get exact solutions of the Schrӧdinger equation. To do this, a real potential is often selected to serve as the driving force of the energy eigenvalues and the eigenfunctions of the Schrӧdinger equation1-3. These state solutions reveal the particle dynamics in non- relativistic quantum mechanics2. Numerous researchers have investigated the bound states of the Schrӧdinger equation using variety of potentials and quantum formalism. Some of these potentials play critical roles in many fields of Physics such as Molecular Physics, Solid State and Chemical Physics4. The Manning-Rosen potential has been studied in-depth and have also been utilized in quantum systems and Yukawa potential, and its classes have been studied in Schrӧdinger formalism5,6. In this work, using the Wentzel, Kramers and Brillouin (WKB) quantum approximation, we shall investigate the bound state solutions of the Schrӧdinger equation using a combination of potentials known as the Inversely Quadratic Yukawa/Attractive Coulomb potential plus a Modified Kratzer potential (IQYCKFP). 2. The WKB Theoretical Approximation In this section, we consider the quasiclassical solution of the Schrӧdinger’s equation for the spherically symmetric potentials. Given the Schrӧdinger equation for a spherically symmetric potentials 𝑉(𝑟) of Equation 3 as (−𝑖ђ)2 ( ∂2 ∂r2 + 1 𝑟2 ∂2 ∂θ2 + 1 𝑟2sin2θ ∂2 ∂ϕ2 ) 𝜓(𝑟, 𝜃, 𝜙) = [2𝑚(𝐸 − 𝑉(𝑟))] 𝜓(𝑟, 𝜃, 𝜙) (1) ABSTRACT: The main objective of this research work is theoretical investigate the bound state solutions of the non- relativistic Schrödinger equation with a mixed potential composed of the Inversely Quadratic Yukawa/Attractive Coulomb potential plus a Modified Kratzer potential (IQYCKFP) by utilizing the Wentzel-Kramers-Brillouin (WKB) quantum theoretical formalism. The energy eigenvalues and its associated wave functions have successfully been obtained in sequel to certain diatomic molecules includes; HCL, HBr, LiH. http://revista.iq.unesp.br/ojs/index.php/ecletica/index https://doi.org/10.26850/1678-4618eqj.v44.3.2019.p50-55 mailto:louismuzong@gmail.com https://orcid.org/0000-0001-8170-0794 https://orcid.org/0000-0002-0286-2865 https://orcid.org/0000-0002-5277-3836 Original article 51 Eclética Química Journal, vol. 44, n. 3, 2019, 50-55 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v44.3.2019.p50-55 The total wave function in Equation 3 can be defined as 𝜓(𝑟, 𝜃, 𝜙) = [𝑟𝑅(𝑟)][√𝑠𝑖𝑛𝜃𝛩(𝜃)𝛷(𝜙)] (2) And by decomposing the spherical wave function in Equation 1 using Equation 2 we obtain the following equations: (−𝑖ђ d dr ) 2 𝑅(𝑟) = [2𝑚(𝐸 − 𝑉(𝑟)) − �⃗⃗� 2 𝑟2 ] 𝑅(𝑟), (3) (−𝑖ђ d d𝜃 ) 2 𝛩(𝜃) = [�⃗⃗� 2 − 𝑀𝑧 2 sin2θ ] 𝛩(𝜃), (4) (−𝑖ђ d d𝜙 ) 2 𝛷(𝜙) = 𝑀𝑧 2𝛷(𝜙) (5) where �⃗⃗� 2, 𝑀𝑧 2 are the constants of separation and, at the same time, integrals of motion. The squared angular momentum �⃗⃗� 2 = (𝑙 + 1 2 ) 2 ђ2. Considering Equation 6, the leading order WKB quantization condition appropriate to Equation 3 is ∫ √𝑃2(𝑟) 𝑟2 𝑟1 𝑑𝑟 = 𝜋ђ(𝑛 + 1 2 ), n=0, 1, 2 (6) where 𝑟2 & 𝑟1 are the classical turning point known as the roots of the equation 𝑃2(𝑟) = 2𝑚(𝐸 − 𝑉(𝑟)) − (𝑙+ 1 2 ) 2 ђ2 𝑟2 = 0 (7) Equation 9 is the WKB quantization condition which is subject for discussion in the preceding section. Consider Equations 5-7 in the framework of the quasi-classical method, the solution of each of these equations in the leading ђ approximation can be written in the form Ψ𝑊𝐾𝐵(𝑟) = 𝐴 √𝑃(𝑟,𝜆) 𝑒𝑥𝑝 [± 𝑖 ђ ∫√𝑃2(𝑟) 𝑑𝑟] (8) 3. Solutions of the Schrödinger Equation The Wentzel, Kramers and Brillouin surmise has been of tremendous importance to physicist, chemist, mathematician as regards quantum mechanics in view of the fact that it gives approximate solutions to linear differential equations. The inversely quadratic Yukawa/attractive Coulomb plus Kratzer Fues potential can be expressed thus 2 0 0 02 1 1 ( ) ( ) (2 ) ( 2 )V r V V A V r r  = − + − + − and 2 ( ) e e r r V r D r −  =     (9) The sum of these potentials can be written as 2 20 0 02 2 2 2 ( ) 2 e e e e e V A V D r D r V r V D r r r r r  = − − − + − + (10) https://doi.org/10.26850/1678-4618eqj.v44.3.2019.p50-55 Original article 52 Eclética Química Journal, vol. 44, n. 3, 2019, 50-55 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v44.3.2019.p50-55 2 2 2 0 0 ( ) 0 2 2 2 2 2 ( 1) 2 2 e e e e eff r e V A D r V D r V D V r r r r r mr   + = − + − − − + + (11) ( )( ) 2 ( )eff rQ r m E V= − (12a) Equation 12a stands for the classical formula for momentum. ( )1( ) ; 2 rb ra Q r dr n = + n=0, 1,2,3… (12b) Upon, substituting Equations 11 and 12a into 12b i.e. the (WKB) we have ( ) 2 2 2 0 0 0 2 2 2 2 2 ( 1) 12 2 22 rb e e e e ne e ra V A D r V D r m E D V dr n r r r r r mr     + − + − + + + − − = +     Factoring out 2m (13) ( ) 2 2 2 0 0 0 2 2 2 2 2 ( 1) 12 2 22 rB e e e e ne e rA V A D r V D r m E D V dr n r r r r r mr     + − + − + + + − − = +     (14) ( ) ( ) 2 2 2 2 0 0 02 1 ( 1) 2 2 2 2 2 1 2 rB ne e e e e e rA m E D V r V A D r r D r V dr r m n      + − + + − + + − − +      = +  (15) ( ) ( ) ( ) 2 2 2 2 0 0 0 1 ( 1) 2 2 2 2 2 1 2 rB ne e e e e e rA m E V D r A D r V r D r V dr r m n      + + − + + − − − +      = +  (16) 2 0 0 2 2 0 2 2 2 where the negative sign on -(A) indicates a bound state. ( 1) 2 ne e e e e e A E V D M A D r V N D r V m    − = + −  = + −   + = − +  Upon substituting the representations made into Equation 16 we have ( ) ( )21 12 . 2 rb ra m Ar Mr N dr n r − + − = + (17) https://doi.org/10.26850/1678-4618eqj.v44.3.2019.p50-55 Original article 53 Eclética Química Journal, vol. 44, n. 3, 2019, 50-55 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v44.3.2019.p50-55 Factoring out A , we have ( )21 12 . 2 rb ra M N mA r r dr n r A A    − + − = +     (18) represent and as M N x y A A ( ) ( )21 12 . 2 rb ra mA r xr y dr n r − + − = + (19) ( )1 12 ( )( ) . 2 rb a b ra mA r r r r dr n r − − = + (20) Where we obtain the classical turning points and a br r from the terms inside the square roots as; 2 4 2 a x x y r − − = , 2 4 2 b x x y r + − = N/N: a b a b r r x r r y + =    =  (21) Recall 1 1 ( )( ) ( ) 2 b a r a b a b a b r r r r r dr r r r r r    − − = + −     (22) ( ) ( )12 2 22 mA x y n   − = + (23) ( ) 2 2 2 14 2 2 mM A n mN =  + +   (24) Upon substituting the coefficients of M, N, A into Equation 24 to obtain the energy eigenvalue. ( ) 2 2 0 0 2 2 0 2 2 ( 2 2 ) 2 2 214 ( 1) 2 e e ne e e e m A D r V E V D mD r mV n   + − − − + =  − + + + +     (25a) Initiating the Langer correction term ( ) ( ) 2 11 2 + → + https://doi.org/10.26850/1678-4618eqj.v44.3.2019.p50-55 Original article 54 Eclética Química Journal, vol. 44, n. 3, 2019, 50-55 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v44.3.2019.p50-55 ( ) ( ) ( ) 2 0 2 2 0 2 2 2 02 ( 2 2 ) 22 21 1 2 2 e e ne e e e m A D r V E D V m n V D r   + − = − −   + + + − −    (25b) The above equation results in the bound state energy spectrum with respect to quantum numbers of a vibrating-rotating diatomic molecule subject to the (IQYCKFP) potential. Thus, its corresponding wave function is given as 2 2 2 2 02 02 1 21 ( ) ( 2 )2 2 2 ( ) e e e ne m mD r V r D E V ne r neR N r e        − + + + −  − − −         = ( ) ( ) ( ) 2 2 2 2 1 0 02 2 21;1 ; 2 2 2 2 e e e ne m m F n D r V D E V r    − + + + − − −       (26) 4. Discussion Having obtained the Energy Eigen Value and its corresponding ( ) using the WKB approach for the Schrödinger equation with the (IQYCKFP), we understood that if we set up parameters 2 00, 0 and eD V A Ze=  = ( ) 2 0 2 2 0 2 2 02 ( 2 ) 22 21 1 2 2 ne m A V E V m n V   − = − −   + + + −    (27) 5. Conclusions It is much easy to show that Equation 19 has resulted to a bound state energy spectrum of a vibrating rotating diatomic molecule subject to the inversely quadratic Yukawa plus attractive coulomb potential. Similarly, if 2 00, 0 and 0eD V A Ze =  = ( ) ( ) ( ) 2 2 2 2 2 2 (2 ) 2 21 1 2 2 e e ne e e e m D r E D m n D r = −   + + + +    (28) Equation 28 results to a bound state energy spectrum subject to Kratzer Fues potential. 6. Acknowledgement The authors are thankful to Professor Benedict I. Ita for the scientific encouragement leading to the success of this manuscript. https://doi.org/10.26850/1678-4618eqj.v44.3.2019.p50-55 Original article 55 Eclética Química Journal, vol. 44, n. 3, 2019, 50-55 ISSN: 1678-4618 DOI: 10.26850/1678-4618eqj.v44.3.2019.p50-55 7. References [1] Antia, A. D, Essien, I. E., Umoren, E. B., Eze, C. C., Approximate solution of the non-relativistic Schrödinger equation with inversely quadratic Yukawa plus Mobius square potential via parametric Nikiforov- Uvarov method, advances in Physics theories and application, 44 (2015) 1-13. https://iiste.org/Journals/index.php/APTA/article/view/ 23029/23549. [2]Ita, B. I., Louis, H., Nzeata-Ibe, N., Ikeuba, A., Ozioma, A. U., Thomas, M. O., Pigweh, A. I., Michael, M. O., Approximate 𝒍-states solutions to the Schrödinger equation with Manning-Rosen plus Hellmann potential via WKB approximation scheme, Sri Lankan Journal of Physics 19 (1) (2018) 37-45. https://doi.org/10.4038/sljp.v19i1.8050. [3] Onate, C. A., Adebimpe, O., Lukman, A. F., Adama, I. J., Okoro, J. O., Davids, E. O., Approximate eigensolutions of the attractive potential via parametric Nikiforov-Uvarov method, Heliyon 4 (11) (2018). https://doi.org/10.1016/j.heliyon.2018.e00977. [4] Hitler, L., Iserom, I. B., Tchoua, P., Ettah, A. A., Bound state solutions of the Klein-Gordon equation for the more general exponential screened Coulomb potential plus Yukawa (MGESCY) potential using Nikiforov-Uvarov method, J. Phys. Math. 9 (1) (2018). https://doi.org/10.4172/2090-0902.1000261. [5] Ikot, A. N., Hassanabadi, H., Maghsoodi, E., Zarrinkamar, S., Relativistic symmetries of Hulthén potential incorporated with generalized tensor interactions, advances in high energy physics 2013 (2013) 1-10. https://doi.org/10.1155/2013/910419. [6] Ita, B. I., Louis, H., Akakuru, O. U., Magu, T. O., Joseph, I., Tchoua, P., Amos, P. I., Effiong, I., Nzeata, N. A., Bound state solutions of the Schrödinger equation for the more general exponential screened Coulomb potential plus Yukawa (MGESCY) potential using Nikiforov-Uvarov method, Journal of Quantum Information Science 8 (1) (2018) 24-45. https://doi.org/10.4236/jqis.2018.81003. https://doi.org/10.26850/1678-4618eqj.v44.3.2019.p50-55 https://iiste.org/Journals/index.php/APTA/article/view/23029/23549 https://iiste.org/Journals/index.php/APTA/article/view/23029/23549 https://iiste.org/Journals/index.php/APTA/article/view/23029/23549 https://iiste.org/Journals/index.php/APTA/article/view/23029/23549 https://iiste.org/Journals/index.php/APTA/article/view/23029/23549 https://iiste.org/Journals/index.php/APTA/article/view/23029/23549 https://iiste.org/Journals/index.php/APTA/article/view/23029/23549 https://iiste.org/Journals/index.php/APTA/article/view/23029/23549 https://doi.org/10.4038/sljp.v19i1.8050 https://doi.org/10.4038/sljp.v19i1.8050 https://doi.org/10.4038/sljp.v19i1.8050 https://doi.org/10.4038/sljp.v19i1.8050 https://doi.org/10.4038/sljp.v19i1.8050 https://doi.org/10.4038/sljp.v19i1.8050 https://doi.org/10.4038/sljp.v19i1.8050 https://doi.org/10.1016/j.heliyon.2018.e00977 https://doi.org/10.1016/j.heliyon.2018.e00977 https://doi.org/10.1016/j.heliyon.2018.e00977 https://doi.org/10.1016/j.heliyon.2018.e00977 https://doi.org/10.1016/j.heliyon.2018.e00977 https://doi.org/10.4172/2090-0902.1000261 https://doi.org/10.4172/2090-0902.1000261 https://doi.org/10.4172/2090-0902.1000261 https://doi.org/10.4172/2090-0902.1000261 https://doi.org/10.4172/2090-0902.1000261 https://doi.org/10.4172/2090-0902.1000261 https://doi.org/10.1155/2013/910419 https://doi.org/10.1155/2013/910419 https://doi.org/10.1155/2013/910419 https://doi.org/10.1155/2013/910419 https://doi.org/10.1155/2013/910419 https://doi.org/10.4236/jqis.2018.81003 https://doi.org/10.4236/jqis.2018.81003 https://doi.org/10.4236/jqis.2018.81003 https://doi.org/10.4236/jqis.2018.81003 https://doi.org/10.4236/jqis.2018.81003 https://doi.org/10.4236/jqis.2018.81003 https://doi.org/10.4236/jqis.2018.81003 https://doi.org/10.4236/jqis.2018.81003